The strengthening effect of grain boundaries will depend upon the temperature of testing. It has been found that above approximately one half of the absolute melting point of the metal, that the grain boundaries are weaker than
the interior of the grain. The temperatures considered in this investigation are in the region where grain boundaries may be expected to strengthen the metal.
There are two main mechanisms responsible for the strengthening effect of grain boundaries (a) the Barrier effect, (b) the complexity effect.
(a.) The Barrier Effect
The grain boundary acts as a barrier to dislocation movement because it is a region of irregular lattice arrange ment and the grains on either side are orientated at an angle to each other. This means that there is no continuity of slip planes across the boundaries. In metals with few slip systems the effect is greater. Thus the barrier effect is greater in h.c.p’. than in b.c.c. or f.c.c. metals. As might be expected the effect is, greater in fine grain metals than in coarse grain metals since the active length of slip plane
is smaller before pile-up at a grain 'boundary occurs. Sub sequent dislocations pile-up "behind the first until suffi cient hack stress is created to stop the source operating.
Eshelby et al. made a calculation relating slip dis
placement to grain size. The hack stress exerted hy a dis location on its source varies inversely as the distance between the two. More dislocations are, therefore, needed near a grain boundary to offset the applied stress at the centre of a large grain than of a small grain. Their calcu lations showed that the number of dislocations n which can be compressed into the distance L between source and
obstacle by a shear stress t is given by the equation:-
n ^
IC is equal to unity for screw dislocations and to
(l - v) where v is Poisson ratio, for edge dislocations.. G- is the shear modulus and b is the Burgers vector. The source will continue to produce dislocations because of the external stress X until the magnitude of the back stress T b is
equal to the applied stress less stress ~C a necessary to activate the source.
X b = T - T a
The number of dislocations emitted by a source located at the centre of a grain of diameter d would be:-
= K i r r a ( .
4 Q-t) ...
Petch^45^ related yield stress and grain size
supposing that yielding takes place when a critical stress (<s^), which is independent of grain size, is reached at the
head of a pile-up of dislocations at the grain boundary. Using the Eshelby, Frank and Nabarro relationship for an edge dislocation:-
-n *
_
TTCr^aCl - V)
_ , ^n<s* - ~ ~ ° * * * * ' '
where n is the number of dislocations in the pile, and is the effective applied stress thus:-
^ = / T f f r ^ F 7 (34)
_JL
The plots of yield stress versus d~2 when extrapolated back to d 2 = 0, indicated that a finite stress was still necessary for yielding. Petch assumed that an average inter nal stress ^ has to be overcome by the applied stress <5"
so that <3"' = - <3“ . Thus substituting in equation (34)
and rearranging:-
I
2Gb cr0er - °"i +^td(l - v) ' • ' -(35^
/ 2Gbcy*
If the value of / v ) eclu-ated to a constant
Ky, then the Petch relationship is established:- _ i_
cTcj = « r i + Kyd 2 . . . .(36)
The slope of the graph obtained when yield stress is plotted against d 2 is equal to Ky and the intercept at d 2 = 0 is equal to <5*^.
The exact nature of the terms ^ and Ky is subject to discussion, however. It is generally accepted that <s~j_ is a friction stress term indicating the resistance of the
as a term indicating the strength of dislocation locking, or the difficulty of nucleating slip in a neighbouring grain.
(b) Complexity Effect
Experimental observation has shown that cavities do not form at the grain boundaries, when a polycrystalline metal is deformed, until test temperatures of the order of half the absolute melting point are reached. Below this temperature it is, therefore, necessary for the polycrystalline aggregate to deform in a complex manner in order to maintain intergran- ular continuity. This effect is known as the complexity
effect and accounts for a large proportion of the strengthen ing due to the presence of grain boundaries. G-.I. Taylor has shown from theoretical considerations that for the grains to conform to each other’s changing shape, individual crystals in a polycrystal must slip on at least five slip systems.
This concept leads to two strengthening mechanisms: (a) When a stress below the yield stress is applied to a polycrystal line metal, it may be sufficient to cause yielding in certain grains which are favourably orientated. However, yielding is restrained until the applied stress is sufficient to activate slip in adjacent grains. Yielding is, therefore, delayed until the whole aggregate can deform together, thus resulting in an increased yield stress.
Because of the complexity effect, individual grains are forced to slip on several slip systems thus hardening by
intersecting slip is more likely, Lomer-Cottrell locking and jog formation being mainly responsible.
near grain boundaries in order to maintain intergranular cohesion means that strain hardening will be most rapid when the grain size is small enough for multiple slip to reach across each grain.
The degree to which any of the above mechanisms is operative in a particular metal is difficult to ascertain. As previously noted in section 2.6. it is also possible to
bring about phase transformation, from 't -*£.-> <*- , by
application of a certain strain which aids the chemical driv ing force involved. It has been observed experimentally that where transformation is preferred to slip that greater elong-
(84-)
ation values are obtainedv . Presumably the more orderly
arrangement of dislocations, produced by the transformation, provides fewer obstacles to further slip than when deforma
tion is wholly slip in character.